Uniform Mordell–Lang conjecture

In a standard uniform formulation: for every choice of complexity bounds for a subvariety $X$ and every integer $r\ge 0$ , there is a constant $C$ such that, for every semiabelian variety $A$ over a field of characteristic $0$ , every subvariety $X\subseteq A$ satisfying those complexity bounds, and every subgroup $\Gamma\subseteq A(\overline{K})$ of rank at most $r$ , the intersection $X\cap\Gamma$ is a union of at most $C$ cosets of groups of the form $\Gamma\cap H$ , where $H$ is an algebraic subgroup of $A$ . The bound $C$ is required to be independent of the ambient semiabelian variety $A$ . The supplied source states the conjecture only by name and claims a proof; it does not specify the precise complexity parameters or the exact bound.

References

Primary source

arXiv

Additional references

Uniform Mordell-Lang conjecture for semiabelian varieties — arXiv — Zhaobo Han, Wenbin Luo, Jiawei Yu

arithmetic-geometry diophantine-geometry

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to prove the uniform Mordell–Lang conjecture for semiabelian varieties, but the proof has not yet been independently verified.

The conjecture seeks uniform bounds and structural control for intersections of subvarieties with finite-rank subgroups in abelian or semiabelian varieties. The curve case traces back to a question of Mazur.

Known results

Dimitrov, Gao, and Habegger, 2021, and Kühne, 2021: the curve case was proved.
David and Philippon: uniform results for subvarieties of self-products of an elliptic curve.
Gao, Ge, and Kühne, 2021, revised 2026: claimed the uniform theorem for abelian varieties, with bounds independent of the ambient variety.

September 2026 claimed semiabelian proof

On September 15, 2026, Zhaobo Han, Wenbin Luo, and Jiawei Yu announced a proof for semiabelian varieties. The retrieved abstract gives no proof details, so this is a claimed resolution rather than a verified theorem.

Current status (as of September 2026): A proof is claimed for the semiabelian case, while independent verification is not recorded; the earlier abelian and curve cases are established in the cited literature.

Sources

No solutions have been posted yet.